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If radius of earth is reduced to half without changing its mass, then match the following columns and mark
the correct option from the codes given below.
Column I
(A) Angular velocity of earth
(B) Time period of rotation of earth
(C) Rotational kinetic energy of earth
Column II
(p) will become two times
(q) will become four times
(r) will remain constant
(s) None

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A B C

r s p

A B C

q r p

A B C

q s q

A B C

q p s

answer is C.

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Detailed Solution

$L_{1} = I ω = \frac{2}{5} M R^{2} ω$

$L_{2} = \frac{2}{5} M {(\frac{R}{2})}^{2} ω^{'}$

As, $L_{1} = L_{2} \Rightarrow ω^{'} = 4 ω$

$∴ \frac{2 π}{T^{'}} = 4 (\frac{2 π}{T}) \Rightarrow T = \frac{T}{4}$

Further, $K = \frac{L^{2}}{2 I}$

Since, angular momentum is constant and I has become (1/4)th.
Therefore, kinetic energy will become 4 times.
Hence, $A \to q, B \to s, C \to q$

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